Simple, semisimple and reductive Lie algebras

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Problem 3.6 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

A Lie algebra LL is called simple if LL is non-abelian and has no proper ideals. Show that for a simple Lie algebra the derived Lie algebra L(1)=[L,L]L^{(1)} = [L,L] is equal to LL, i.e.L(1)=LL^{(1)} = L.

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Problem 3.7 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

A Lie algebra LL is called semisimple if L≠0L \neq 0 and LL has no abelian ideals ≠0\neq 0. Show that a finite dimensional Lie algebra LL is semisimple if and only if rad⁡L=0\operatorname {rad}L = 0.

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Problem 3.8 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Show that the adjoint representation of a semisimple Lie algebra is faithful. A faithful representation of a Lie algebra is a linear representation in which different elements of LL are represented by distinct linear mappings.

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Problem 3.12 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

The Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) is spanned by the 2×22 \times 2 matrices

h=(100−1),e=(0100),f=(0010) h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \qquad e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}

with

[e,f]=h,[h,e]=2e,[h,f]=−2f . [e,f] = h, \qquad [h,e] = 2e, \qquad [h,f] = -2f \, .
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(i)

Let xx be a linear combination of the basis h,e,fh, e, f

x=c1h+c2e+c3f x = c_{1}h + c_{2}e + c_{3}f

where c1,c2,c3∈Rc_{1}, c_{2}, c_{3} \in \mathbb {R}. Calculate the commutators [h,x],[e,x],[f,x][h,x], [e,x], [f,x].

(ii)

Show that the Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) is simple.

Problem 3.13 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

The Lie algebra sl(2,R)\mathfrak {sl}(2,\mathbb {R}) is simple. Study the Lie algebra sl(2,F)\mathfrak {sl}(2,\mathbb {F}), where char⁡F=2\operatorname {char}\mathbb {F} = 2.

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Problem 3.14 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Show that for a simple Lie algebra LL the derived algebra L(1)=[L,L]L^{(1)} = [L,L] is equal to LL.

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Problem 3.17 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Give a Lie algebra which is semisimple, but not simple. Provide a matrix representation.

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Problem 3.21 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

A finite-dimensional Lie algebra over a field F\mathbb {F} of characteristic 0 (for example R\mathbb {R} and C\mathbb {C}) is called reductive if its adjoint representation is completely reducible. Give an example of a Lie algebra which is reductive but not semisimple.

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Problem 3.30 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Let XX be a 2×22 \times 2 matrix over C\mathbb {C}. Let

L=(−1001) . L = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \, .
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(i)

Find the condition on XX such that X∗L+LX=02X^{*}L + LX = 0_{2}. Then XX is an element of the Lie algebra u(1,1)\mathfrak {u}(1,1).

(ii)

Give a basis of the Lie algebra. Is the Lie algebra semisimple?

Problem 3.33 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Let e1,e2,e3e_{1}, e_{2}, e_{3} be the generators of a three-dimensional Lie algebra with the commutation relation

[e1,e3]=e1,[e2,e3]=e1+e2,[e1,e2]=0 . [e_{1},e_{3}] = e_{1}, \qquad [e_{2},e_{3}] = e_{1}+e_{2}, \qquad [e_{1},e_{2}] = 0 \, .

Find the adjoint representation. Is the Lie algebra semisimple?

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Problem 3.64 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Consider the Lie algebra so(4)\mathfrak {so}(4). Show that the Lie algebra so(4)\mathfrak {so}(4) is semisimple, but not simple. Consider as a basis of so(4)\mathfrak {so}(4) the six vector fields

V12=x2∂∂x1−x1∂∂x2,V23=x3∂∂x2−x2∂∂x3,V31=x1∂∂x3−x3∂∂x1 V_{12} = x_{2}\frac{\partial }{\partial x_{1}} - x_{1}\frac{\partial }{\partial x_{2}}, \qquad V_{23} = x_{3}\frac{\partial }{\partial x_{2}} - x_{2}\frac{\partial }{\partial x_{3}}, \qquad V_{31} = x_{1}\frac{\partial }{\partial x_{3}} - x_{3}\frac{\partial }{\partial x_{1}} W41=x1∂∂x4−x4∂∂x1,W42=x2∂∂x4−x4∂∂x2,W43=x3∂∂x4−x4∂∂x3 . W_{41} = x_{1}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{1}}, \qquad W_{42} = x_{2}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{2}}, \qquad W_{43} = x_{3}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{3}} \, .
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Problem 3.72 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"
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(i)

Consider the matrices

N1=(00i000000),N2=(00000i000),L=(0−i/20i/200000) . N_{1} = \begin{pmatrix} 0 & 0 & i \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad N_{2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & i \\ 0 & 0 & 0 \end{pmatrix}, \qquad L = \begin{pmatrix} 0 & -i/2 & 0 \\ i/2 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \, .

Show that they form a basis of a Lie algebra.

(ii)

Is the Lie algebra semisimple?

Problem 3.79 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Consider the 4×44 \times 4 matrices

J+=(0000001000000000),J−=(0100000000000000), J_{+} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad J_{-} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, J3=(0001010000000000),E=(0010000000000000) . J_{3} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad E = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \, .
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(i)

Find the commutators and thus show that we have a basis of a Lie algebra. Is the Lie algebra semisimple?

(ii)

Let α,β,γ,δ∈R\alpha , \beta , \gamma , \delta \in \mathbb {R}. Calculate

exp(αJ+),exp(βJ−),exp(γJ3),exp(δE) . \mathrm{exp}\left(\alpha J_{+}\right), \quad \mathrm{exp}\left(\beta J_{-}\right), \quad \mathrm{exp}\left(\gamma J_{3}\right), \quad \mathrm{exp}\left(\delta E\right) \, .

Show that exp(αJ+)exp(δE)=exp(δE)exp(αJ+)\mathrm{exp}\left(\alpha J_{+}\right)\mathrm{exp}\left(\delta E\right) = \mathrm{exp}\left(\delta E\right)\mathrm{exp}\left(\alpha J_{+}\right).

Problem 3.88 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

Let c,x∈Rc, x \in \mathbb {R}. Consider the operators

K1=−iccos⁡(x),K2=−icsin⁡(x),K3=−ddx . K_{1} = -ic\cos (x), \qquad K_{2} = -ic\sin (x), \qquad K_{3} = -\frac{d}{dx} \, .

Find the commutators and thus show that we have a basis of a Lie algebra. Is the Lie algebra semisimple?

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Problem 3.92 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

The Euclidean group E(2)E(2) in the plane is associated with the Lie algebra e(2)\mathfrak {e}(2) defined by the commutation relations

[X1,X2]=X3,[X1,X3]=−X2,[X2,X3]=0 . [X_{1},X_{2}] = X_{3}, \qquad [X_{1},X_{3}] = -X_{2}, \qquad [X_{2},X_{3}] = 0 \, .

Show that the Lie algebra is not semisimple.

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Problem 3.110 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Section 3.2 "Supplementary Problems"

The infinitesimal generators (vector fields) of the six dimensional Lie algebra so(4)\mathfrak {so}(4) can be written in terms of the variables (x1,x2,x3,x4)(x_{1},x_{2},x_{3},x_{4}) as

X1=x3∂∂x2−x2∂∂x3,X2=x1∂∂x3−x3∂∂x1,X3=x2∂∂x1−x1∂∂x2, X_{1} = x_{3}\frac{\partial }{\partial x_{2}} - x_{2}\frac{\partial }{\partial x_{3}}, \qquad X_{2} = x_{1}\frac{\partial }{\partial x_{3}} - x_{3}\frac{\partial }{\partial x_{1}}, \qquad X_{3} = x_{2}\frac{\partial }{\partial x_{1}} - x_{1}\frac{\partial }{\partial x_{2}}, Y1=x1∂∂x4−x4∂∂x1,Y2=x2∂∂x4−x4∂∂x2,Y3=x3∂∂x4−x4∂∂x3 . Y_{1} = x_{1}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{1}}, \qquad Y_{2} = x_{2}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{2}}, \qquad Y_{3} = x_{3}\frac{\partial }{\partial x_{4}} - x_{4}\frac{\partial }{\partial x_{3}} \, .
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(i)

Find the commutators.

(ii)

Show that the Lie algebra is semisimple, but not simple.

(iii)

Find elements of the compact Lie group SO(4)SO(4) by calculating

exp(α1X1+α2X2+α3X3+β1Y1+β2Y2+β3Y3) \mathrm{exp}\left(\alpha _{1}X_{1}+\alpha _{2}X_{2}+\alpha _{3}X_{3}+\beta _{1}Y_{1}+\beta _{2}Y_{2}+\beta _{3}Y_{3}\right)

and

exp(α1X1)exp(α2X2)exp(α3X3)exp(β1Y1)exp(β2Y2)exp(β3Y3) \mathrm{exp}\left(\alpha _{1}X_{1}\right)\mathrm{exp}\left(\alpha _{2}X_{2}\right)\mathrm{exp}\left(\alpha _{3}X_{3}\right)\mathrm{exp}\left(\beta _{1}Y_{1}\right)\mathrm{exp}\left(\beta _{2}Y_{2}\right)\mathrm{exp}\left(\beta _{3}Y_{3}\right)

where αj,βj∈R\alpha_{j}, \beta_{j} \in \mathbb {R}.

References

[STH12] Steeb W., Tanski I., Hardy Y. Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. World Scientific Publishing, 2012.